\chapter{Gaussian laws conditioned on a sphere}\label{sec:conditioning}
\label{sec-common-hot-tools}%

This section and the next collect deterministic estimates for a centered Gaussian vector with independent coordinates, conditioned on the event that its squared norm equals its expectation.
We show in \cref{prop:cond-bingham} that the Gibbs measures $\pi_A$ of \cref{sec:model} are laws of this kind, and so are the Bingham comparison laws $\mu_t$ of \cref{sec:heat}.
The main results are a quantitative local limit theorem for weighted sums of centered squares of Gaussian variables (\cref{gp-local-limit}); a formula for the density of a set of coordinates under the conditioned law, relative to independent Gaussian variables, together with a comparison when these coordinates carry a small fraction of the variance (\cref{gp-head-density,gp-growing-head}); moment bounds (\cref{gp-moments}); an exact formula for quadratic exponential moments (\cref{gp-real-tilt}); and moment bounds for quadratic statistics (\cref{gp-quadratic-lp,gp-quadratic-variance}).
We treat finite and countably infinite variance sequences together.
The countable case is needed in the critical window, where the limiting Gibbs law is a countable Gaussian sequence conditioned on a renormalized radial constraint.
We construct this law in \cref{lem:cond-consistent}, identify its finite-coordinate marginals in \cref{gp-head-density}, and prove in \cref{prop:cond-continuity} that it depends continuously on the variance sequence, with moment bounds that hold uniformly along a converging sequence of finite approximations; \cref{sec:crit-process} specializes these results to the critical variances.

The proofs use only the notation of \cref{sec:model,sec:spectral}.
The results are used in \cref{sec:likelihood}, in the comparison and moment bounds of \cref{sec:comparison,sec:process,sec:density}, in \cref{part:hot}, and in \cref{sec:crit-process}.
All statements are deterministic.
A constant is called universal if it does not depend on the number of coordinates, on the size of the smallest variance, or on any other feature of the variance sequence.

\section{Notation}\label{sec:cond-notation}

A \emph{variance sequence} is a family $v=(v_i)_{i\in I}$ of positive real numbers, indexed by a nonempty finite or countably infinite set $I$, such that $\sum_{i\in I}v_i^2<\infty$.
We call $v$ finite or countable according to whether $I$ is finite or countably infinite.
Let $g=(g_i)_{i\in I}$ be independent standard Gaussian random variables, and define
\begin{equation}\label{gp-standardization}
T_v=\sum_{i\in I}v_i(g_i^2-1),\qquad
S_v=\Big(2\sum_{i\in I}v_i^2\Big)^{1/2},\qquad
\hat v_i=\frac{v_i}{S_v},\qquad
\eta_v=\max_{i\in I}\hat v_i .
\end{equation}
For countable $I$, the summands of $T_v$ are independent and centered, with variances $2v_i^2$ whose sum is finite.
The series converges in $L^2$ to a limit that does not depend on the order of summation.
For any enumeration of $I$, its partial sums form a martingale bounded in $L^2$, which converges almost surely by the martingale convergence theorem.
We have $\Var T_v=S_v^2$ and $\sum_i\hat v_i^2=1/2$, so that $\eta_v\le2^{-1/2}$; the maximum in \eqref{gp-standardization} is attained because $\hat v_i\to0$ along any enumeration of a countable $I$.
The quantity $\eta_v$ is small exactly when no single coordinate carries a macroscopic part of the variance of $T_v$.
For instance, if $v_i=1$ for $1\le i\le n$, then $S_v=(2n)^{1/2}$ and $\eta_v=(2n)^{-1/2}$.

When $T_v$ has a continuous density, we denote it by $p_v$, and write $f_v(x)=S_vp_v(S_vx)$ for the density of $T_v/S_v$.
We write $\hat f_v(t)=\E e^{\mathrm itT_v/S_v}$ for the characteristic function of $T_v/S_v$.
The standard Gaussian density and distribution function are $\phi$ and $\Phi$, and $G$ denotes a standard Gaussian random variable.

For $A\subset I$, we write $v|_A$ for the restriction of $v$ to $A$, and $T_{v,A}=\sum_{i\in A}v_i(g_i^2-1)$ and $p_{v,A}$ for the corresponding sum and its density.
We let $\gamma_A$ be the standard Gaussian law on $\R^A$, the product of standard Gaussian laws on the coordinates, and, for finite $A$, $\phi_A(z)=\prod_{i\in A}\phi(z_i)$ its density.
For $z\in\R^A$ we set
\begin{equation}\label{gp-head-quantities}
U_A(z)=\sum_{i\in A}v_i(z_i^2-1),\qquad
h_A=\frac{\sum_{i\in A}v_i^2}{\sum_{i\in I}v_i^2},\qquad
S_{v|_{I\setminus A}}=S_v(1-h_A)^{1/2}.
\end{equation}
For infinite $A$, the series $U_A(z)$ is defined for $\gamma_A$-almost every $z$, as the almost sure and $L^2(\gamma_A)$ limit of its partial sums.
Note that $T_{v,A}=U_A(g_A)$, where $g_A=(g_i)_{i\in A}$, and $h_A$ is the fraction of the variance of $T_v$ contributed by the coordinates in $A$.

Under the conditioned laws defined below, the random point is denoted by $Y=(Y_i)_{i\in I}$, and its standardized coordinates are $Z_i=Y_i/\sqrt{v_i}$.
For a probability measure $\mu$ and a coordinate set $A$, we write $\mu^{Z_A}$ for the law of $Z_A=(Z_i)_{i\in A}$ under $\mu$.
For $r>0$, let $\mathcal S_r=\{y\in\R^I:|y|^2=r\}$ when $I$ is finite, with $(|I|-1)$-dimensional surface measure $\mathcal H^{|I|-1}$; in particular $\mathcal S_N=S_N$ when $I=\{1,\dots,N\}$.

\section{Densities of weighted sums of centered squares}\label{sec:cond-densities}

The following lemma is the basis of all density estimates in this section.

\begin{lemma}[A differential equation for the characteristic function]\label{gp-scalar-ode}
Let $v$ be a variance sequence.
The characteristic function $\hat f_v$ is continuously differentiable and satisfies
\begin{equation}\label{gp-scalar-ode-equation}
\hat f_v'(t)=\omega_v(t)\hat f_v(t),\qquad
\omega_v(t)=-2t\sum_{i\in I}\frac{\hat v_i^2}{1-2\mathrm it\hat v_i},\qquad
\hat f_v(0)=1,
\end{equation}
where the series defining $\omega_v$ converges uniformly on bounded intervals.
Further,
\begin{equation}\label{gp-fourier-envelope}
|\hat f_v(t)|=\prod_{i\in I}\big(1+4t^2\hat v_i^2\big)^{-1/4}
\le\big(1+4\eta_v^2t^2\big)^{-1/(8\eta_v^2)},
\end{equation}
and, if $|t|\eta_v\le1/4$, then
\begin{equation}\label{gp-fourier-central-error}
\big|\hat f_v(t)-e^{-t^2/2}\big|\le\eta_v|t|^3e^{-2t^2/5}.
\end{equation}
\end{lemma}

\begin{proof}
We begin with one coordinate.
For $b>0$, let $k_b(t)=\E e^{\mathrm itb(G^2-1)}$.
We may differentiate under the expectation, since $\E G^2<\infty$.
Applying Gaussian integration by parts, $\E[G\psi(G)]=\E[\psi'(G)]$, to $\psi(x)=xe^{\mathrm itbx^2}$, we find that
\[
\E\big[G^2e^{\mathrm itbG^2}\big]=\E\big[e^{\mathrm itbG^2}\big]+2\mathrm itb\,\E\big[G^2e^{\mathrm itbG^2}\big],
\quad\text{that is,}\quad
(1-2\mathrm itb)\,\E\big[G^2e^{\mathrm itbG^2}\big]=\E\big[e^{\mathrm itbG^2}\big].
\]
Inserting this into $k_b'(t)=\mathrm ib\,e^{-\mathrm itb}\big(\E[G^2e^{\mathrm itbG^2}]-\E[e^{\mathrm itbG^2}]\big)$, we obtain
\[
k_b'(t)=-\frac{2tb^2}{1-2\mathrm itb}\,k_b(t).
\]
Taking the real part of $2\overline{k_b}k_b'$, we find that $|k_b|^2$ solves the real equation
\[
\frac{\dd}{\dd t}|k_b(t)|^2=-\frac{4tb^2}{1+4t^2b^2}|k_b(t)|^2,
\]
and solving this equation with $|k_b(0)|=1$ gives $|k_b(t)|=(1+4t^2b^2)^{-1/4}$.
For finite $I$, the variable $T_v/S_v$ is the sum of the independent variables $\hat v_i(g_i^2-1)$, and $\hat f_v=\prod_ik_{\hat v_i}$.
Differentiating this finite product yields \eqref{gp-scalar-ode-equation} and the identity in \eqref{gp-fourier-envelope}.

Let $I$ be countable, fix an enumeration of $I$, and let $X_n$ be the $n$-th partial sum of the series $X=T_v/S_v$.
The characteristic functions $\hat f_n$ of $X_n$ converge to $\hat f_v$ uniformly on bounded intervals, because $X_n\to X$ in $L^2$.
Their derivatives $\hat f_n'(t)=\mathrm i\E[X_ne^{\mathrm itX_n}]$ also converge uniformly for $|t|\le T$, since
\[
\begin{split}
\big|\E[X_ne^{\mathrm itX_n}]-\E[Xe^{\mathrm itX}]\big|
&\le\E|X_n-X|+\E\big[|X|\,|e^{\mathrm itX_n}-e^{\mathrm itX}|\big]\\
&\le\|X_n-X\|_2+T\|X\|_2\|X_n-X\|_2 .
\end{split}
\]
The summands of $\omega_v$ have modulus at most $2T\hat v_i^2$ for $|t|\le T$, and the coefficients of the finite equations converge to $\omega_v$ uniformly for $|t|\le T$.
Passing to the limit in the finite equations, we find \eqref{gp-scalar-ode-equation}; the finite products of moduli converge to the infinite product in \eqref{gp-fourier-envelope}.

For the inequality in \eqref{gp-fourier-envelope}, we use that $a\mapsto a^{-1}\log(1+ax)$ is nonincreasing on $a>0$ for every $x\ge0$, since its derivative has the sign of $ax/(1+ax)-\log(1+ax)\le0$.
Using $\hat v_i^2\le\eta_v^2$, we find
\[
\sum_{i\in I}\log\big(1+4t^2\hat v_i^2\big)
\ge\sum_{i\in I}\frac{\hat v_i^2}{\eta_v^2}\log\big(1+4\eta_v^2t^2\big)
=\frac{1}{2\eta_v^2}\log\big(1+4\eta_v^2t^2\big),
\]
which is the inequality in \eqref{gp-fourier-envelope}.
Next, using $\log(1+x)\ge x/(1+x)$, we have
\[
|\hat f_v(t)|\le\exp\Big\{-\sum_{i\in I}\frac{t^2\hat v_i^2}{1+4t^2\hat v_i^2}\Big\}
\le\exp\Big\{-\frac{t^2}{2(1+4\eta_v^2t^2)}\Big\}
\le e^{-2t^2/5},\qquad |t|\eta_v\le1/4 .
\]
Since $\sum_i\hat v_i^2=1/2$, we have
\[
|\omega_v(t)+t|=\Big|2t\sum_{i\in I}\hat v_i^2\,\frac{2\mathrm it\hat v_i}{1-2\mathrm it\hat v_i}\Big|
\le4t^2\sum_{i\in I}\hat v_i^3\le2\eta_vt^2 .
\]
By variation of constants in \eqref{gp-scalar-ode-equation}, we have, for $t\ge0$,
\[
\hat f_v(t)-e^{-t^2/2}=\int_0^te^{-(t^2-s^2)/2}\big(\omega_v(s)+s\big)\hat f_v(s)\dd s .
\]
For $0\le s\le t\le(4\eta_v)^{-1}$, the integrand has modulus at most $2\eta_vs^2e^{-t^2/2+s^2/2-2s^2/5}\le2\eta_vs^2e^{-2t^2/5}$.
Integrating in $s$ gives \eqref{gp-fourier-central-error} for $t\ge0$.
Negative $t$ follow by complex conjugation.
\end{proof}

\begin{lemma}[Bounded densities from three coordinates]\label{lem:cond-three}
Let $v$ be a variance sequence such that $I$ has at least three elements.
\begin{enumerate}
\item The variable $T_v$ has a bounded continuous density $p_v$.
For any three distinct indices $j_1,j_2,j_3\in I$,
\begin{equation}\label{eq:cond-three-sup}
\|p_v\|_\infty\le\frac{c_3}{\min\{v_{j_1},v_{j_2},v_{j_3}\}},\qquad
c_3=\frac1{4\pi}\int_{\R}(1+s^2)^{-3/4}\dd s .
\end{equation}
\item We have $p_v(t)>0$ for every $t>-\sum_iv_i$, where the sum may be infinite.
If $\sum_iv_i<\infty$, then $p_v(t)=0$ for $t\le-\sum_iv_i$.
\item If $A\subset I$ and $I\setminus A$ has at least three elements, then
\begin{equation}\label{eq:cond-convolution}
p_v(t)=\E\,p_{v,I\setminus A}\big(t-T_{v,A}\big)\qquad\text{for every }t\in\R .
\end{equation}
\end{enumerate}
\end{lemma}

\begin{proof}
By \eqref{gp-fourier-envelope} with $t=uS_v$, we have $|\E e^{\mathrm iuT_v}|=\prod_i(1+4u^2v_i^2)^{-1/4}$.
Every factor is at most one.
Keeping only the three factors with indices $j_1,j_2,j_3$, we have $|\E e^{\mathrm iuT_v}|\le(1+4u^2m^2)^{-3/4}$, where $m=\min\{v_{j_1},v_{j_2},v_{j_3}\}$.
This bound is integrable in $u$, and Fourier inversion yields a bounded continuous density $p_v$ with $\|p_v\|_\infty\le(2\pi)^{-1}\int(1+4u^2m^2)^{-3/4}\dd u=c_3/m$, which proves the first part.

For the third part, $T_v=T_{v,A}+T_{v,I\setminus A}$ with independent summands.
By Fubini's theorem, the right side of \eqref{eq:cond-convolution} is a density of $T_v$, and it is continuous in $t$ by dominated convergence, since $p_{v,I\setminus A}$ is bounded and continuous.
Two continuous densities of the same law agree everywhere, and \eqref{eq:cond-convolution} follows.

For the second part, if $\sum_iv_i<\infty$, then $T_v\ge-\sum_iv_i$ and $p_v$ vanishes on $(-\infty,-\sum_iv_i)$, and by continuity also at $-\sum_iv_i$.
Now let $t>-\sum_iv_i$, and choose a finite set $B\subset I$ with at least three elements and $\sum_{i\in B}v_i>-t$.
By \cite[Lemma~2.2]{DuHuang2026}, the variable $\sum_{i\in B}v_ig_i^2$ has a continuous density that is positive on $(0,\infty)$.
Equivalently, $p_{v,B}$ is positive on $(-\sum_{i\in B}v_i,\infty)$.
By \eqref{eq:cond-convolution} with $A=I\setminus B$, we have
\[
p_v(t)=\E\,p_{v,B}\big(t-T_{v,I\setminus B}\big)
\ge\E\big[p_{v,B}\big(t-T_{v,I\setminus B}\big)\ind\{T_{v,I\setminus B}\le0\}\big]>0 .
\]
In the last step, we used that $t-T_{v,I\setminus B}\ge t>-\sum_{i\in B}v_i$ on the event $\{T_{v,I\setminus B}\le0\}$, and that this event has positive probability because $T_{v,I\setminus B}$ has mean zero (it is identically zero if $B=I$).
\end{proof}

\begin{theorem}[A quantitative local limit theorem]\label{gp-local-limit}
For each integer $j\ge0$, there exist universal constants $\eta_j>0$ and $C_j<\infty$ with the following property.
If $v$ is a variance sequence with $\eta_v\le\eta_j$, then $f_v$ has $j$ continuous derivatives and
\begin{equation}\label{gp-local-limit-bound}
\sup_{x\in\R}\big|f_v^{(j)}(x)-\phi^{(j)}(x)\big|\le C_j\eta_v .
\end{equation}
Further, there exist universal constants $\eta_{\mathrm{loc}},c_0>0$ and $C<\infty$ such that $\eta_v\le\eta_{\mathrm{loc}}$ implies
\begin{equation}\label{gp-local-limit-consequences}
f_v(0)\ge c_0,\qquad \|f_v\|_\infty+\|f_v''\|_\infty\le C,\qquad
|f_v'(0)|\le C\eta_v,\qquad \int_{\R}t^2|\hat f_v(t)|\dd t\le C .
\end{equation}
\end{theorem}

The constants do not depend on the number of coordinates, the smallest variance, or the relative sizes of the variances, and the theorem applies to finite and countable sequences alike.
Note that $\eta_v\le\eta_j$ forces $I$ to have at least $(2\eta_j^2)^{-1}$ elements.

\begin{proof}
Fix $j\ge0$ and let $\eta_j^2\le a_j$, where $a_j=1/(8(j+4))$.
The function $a\mapsto(1+4at^2)^{-1/(8a)}$ is nondecreasing on $a>0$, by the monotonicity used in the proof of \cref{gp-scalar-ode}.
Then \eqref{gp-fourier-envelope} implies that, for $\eta_v\le\eta_j$,
\[
|\hat f_v(t)|\le(1+4a_jt^2)^{-1/(8a_j)}=(1+4a_jt^2)^{-(j+4)},
\]
and $|t|^j|\hat f_v(t)|$ is integrable.
By Fourier inversion, $f_v$ exists, is $j$ times continuously differentiable, and
\[
f_v^{(j)}(x)-\phi^{(j)}(x)=\frac1{2\pi}\int_{\R}(-\mathrm it)^je^{-\mathrm itx}\big(\hat f_v(t)-e^{-t^2/2}\big)\dd t .
\]
We bound the integral of $|t|^j|\hat f_v(t)-e^{-t^2/2}|$ over two regions.
On $|t|\le(4\eta_v)^{-1}$, \eqref{gp-fourier-central-error} bounds it by $\eta_v\int|t|^{j+3}e^{-2t^2/5}\dd t=C_j\eta_v$.
On $|t|>(4\eta_v)^{-1}$, we set $u=2\eta_v|t|\ge1/2$ and split the exponent in \eqref{gp-fourier-envelope} into two equal halves:
\[
|\hat f_v(t)|\le(1+u^2)^{-1/(16\eta_v^2)}(1+u^2)^{-1/(16\eta_v^2)}
\le e^{-\log(5/4)/(16\eta_v^2)}(1+u^2)^{-(j+4)/2},
\]
where we used $u^2\ge1/4$ in the first factor and $1/(16\eta_v^2)\ge(j+4)/2$ in the second.
Changing variables, we find
\[
\int_{|t|>(4\eta_v)^{-1}}|t|^j|\hat f_v(t)|\dd t
\le e^{-\log(5/4)/(16\eta_v^2)}(2\eta_v)^{-j-1}\int_{\R}|u|^j(1+u^2)^{-(j+4)/2}\dd u
\le C_j\eta_v,
\]
since $x^{-j-2}e^{-\log(5/4)/(16x^2)}$ is bounded on $x>0$.
The Gaussian tail $\int_{|t|>(4\eta_v)^{-1}}|t|^je^{-t^2/2}\dd t$ is also at most $C_j\eta_v$.
This proves \eqref{gp-local-limit-bound}, and also $\int|t|^j|\hat f_v(t)-e^{-t^2/2}|\dd t\le C_j\eta_v$.

For \eqref{gp-local-limit-consequences}, let $\eta_{\mathrm{loc}}\le\min\{\eta_0,\eta_1,\eta_2\}$ be so small that $C_0\eta_{\mathrm{loc}}\le\phi(0)/2$.
Using \eqref{gp-local-limit-bound} with $j=0,1,2$ and $\phi'(0)=0$, we have $f_v(0)\ge\phi(0)/2$, the bounds on $\|f_v\|_\infty$ and $\|f_v''\|_\infty$, and $|f_v'(0)|\le C_1\eta_v$.
Finally, $\int t^2|\hat f_v|\le\int t^2e^{-t^2/2}\dd t+C_2\eta_v\le C$ by the last bound of the previous paragraph.
\end{proof}

\section{Conditioning on the sphere: finite sequences}\label{sec:cond-finite}

\begin{definition}[Gaussian law conditioned on a sphere]\label{def:cond-finite}
Let $v$ be a finite variance sequence and $R^2=\sum_{i\in I}v_i$.
The \emph{Gaussian law $\mu_v$ conditioned on the sphere} is the probability measure on $\mathcal S_{R^2}$ given by
\begin{equation}\label{gp-pin-definition}
\mu_v(\dd y)=\frac1{\mathcal Z_v}\exp\Big\{-\frac12\sum_{i\in I}\frac{y_i^2}{v_i}\Big\}\,\mathcal H^{|I|-1}(\dd y),
\qquad y\in\mathcal S_{R^2},
\end{equation}
where $\mathcal Z_v$ is the normalizing constant.
\end{definition}

This is a Bingham distribution \cite{Bingham1974}: a probability measure on a sphere whose density with respect to surface measure is the exponential of a quadratic form.
The next lemma justifies the name: $\mu_v$ is the law of $Y=(\sqrt{v_i}\,g_i)_{i\in I}$ conditioned on $|Y|^2=R^2$, that is, on $T_v=0$.
This description is classical; see \cite[Abstract]{KumeWood2005} for its use in the computation of Bingham normalizing constants, \cite[Section~1.2, (1.7)--(1.8)]{DuHuang2026} for the case of the spherical Sherrington--Kirkpatrick model at the critical temperature, and \cite[Proposition~2.7]{AruKorzhenkova2024} for a lattice spherical model.

\begin{lemma}[Radial densities]\label{lem:cond-radial}
Let $v$ be a finite variance sequence, $n=|I|$, $R^2=\sum_iv_i$, and $Y=(\sqrt{v_i}\,g_i)_{i\in I}$, with density $\gamma_v(y)=\prod_i(2\pi v_i)^{-1/2}e^{-y_i^2/(2v_i)}$.
For a bounded nonnegative measurable function $F$, or a continuous real-valued function $F$, on $\R^I$, and for $t>-R^2$, let
\begin{equation}\label{eq:cond-weighted-density}
p_v[F](t)=\frac{1}{2(R^2+t)^{1/2}}\int_{\mathcal S_{R^2+t}}F(y)\gamma_v(y)\,\mathcal H^{n-1}(\dd y).
\end{equation}
\begin{enumerate}
\item For bounded nonnegative measurable $F$ on $\R^I$ and nonnegative measurable $K$ on $\R$, we have $\E[F(Y)K(T_v)]=\int_{-R^2}^\infty K(t)\,p_v[F](t)\dd t$.
\item If $F$ is continuous, then $p_v[F]$ is finite and continuous on $(-R^2,\infty)$.
In particular, $p_v[1]$ is a continuous and positive version of the density of $T_v$ on $(-R^2,\infty)$.
\item For every continuous $F$, we have $\mu_v(F)=p_v[F](0)/p_v[1](0)$.
For every bounded continuous $F$, we have $\E[F(Y)\,|\,|T_v|\le\eps]\to\mu_v(F)$ as $\eps\downarrow0$.
\end{enumerate}
\end{lemma}

\begin{proof}
In polar coordinates, $\int_{\R^n}H(y)\dd y=\int_0^\infty\int_{\mathcal S_{r^2}}H\,\dd\mathcal H^{n-1}\dd r$ for nonnegative measurable $H$.
Substituting $s=r^2$, we find that $\int_{\R^n}H(y)\dd y=\int_0^\infty(2\sqrt s)^{-1}\int_{\mathcal S_s}H\,\dd\mathcal H^{n-1}\dd s$.
With $H(y)=F(y)\gamma_v(y)K(|y|^2-R^2)$ and $t=s-R^2$, this is the first part, since $T_v=|Y|^2-R^2$.
For the second part, by scaling the sphere, we have
\[
\int_{\mathcal S_{R^2+t}}F\gamma_v\,\dd\mathcal H^{n-1}
=(R^2+t)^{(n-1)/2}\int_{\mathcal S_1}F\big((R^2+t)^{1/2}\theta\big)\gamma_v\big((R^2+t)^{1/2}\theta\big)\,\mathcal H^{n-1}(\dd\theta).
\]
The integrand is jointly continuous in $(t,\theta)$ and the unit sphere is compact, so the integral is continuous in $t>-R^2$.
Taking $F=1$ in the first part, we see that $p_v[1]$ is a density of $T_v$; it is positive because $\gamma_v>0$.

For the third part, recall from \eqref{gp-pin-definition} that the density of $\mu_v$ with respect to surface measure is proportional to $\gamma_v$ on $\mathcal S_{R^2}$, which is the identity $\mu_v(F)=p_v[F](0)/p_v[1](0)$.
Finally, for bounded continuous $F$, the second part implies that
\[
\E\big[F(Y)\,\big|\,|T_v|\le\eps\big]=\frac{\int_{-\eps}^\eps p_v[F](t)\dd t}{\int_{-\eps}^\eps p_v[1](t)\dd t}
\longrightarrow\frac{p_v[F](0)}{p_v[1](0)}=\mu_v(F),
\]
as $\eps\downarrow0$.
\end{proof}

For a nonnegative continuous $F$ that is unbounded on $\R^I$, bounded radial truncation justifies each later use of the coarea identity: on any compact interval $J\subset(-R^2,\infty)$, multiply $F$ by a continuous compactly supported function of $|y|^2$ that equals one on the radii $R^2+J$. The resulting test is bounded and agrees with $F$ on the relevant spheres. Thus the first part applies on $J$; increasing such intervals gives any nonnegative integral identity used below. The exact-level identity already applies directly to continuous polynomial and exponential tests.

For every finite variance sequence, we define $p_v(t)=p_v[1](t)$ for $t>-R^2$ and $p_v(t)=0$ for $t\le-R^2$; this is a density of $T_v$.
When $I$ has at least three elements, it agrees everywhere with the continuous density of \cref{lem:cond-three}, since both are densities of $T_v$ that are continuous on $(-R^2,\infty)$ and vanish on $(-\infty,-R^2]$.
When $I$ has one or two elements, $p_v$ is continuous only on $(-R^2,\infty)$: for one variance $v_1$, it equals $(2\pi v_1(t+v_1))^{-1/2}e^{-(t+v_1)/(2v_1)}$ for $t>-v_1$, which is unbounded near $-v_1$.
Taking $F=1$ in \eqref{eq:cond-weighted-density} at $t=0$ gives the normalization
\begin{equation}\label{eq:cond-normalization}
\int_{\mathcal S_{R^2}}\gamma_v\,\dd\mathcal H^{|I|-1}=2R\,p_v(0),\qquad
\frac{\dd\mu_v}{\dd\mathcal H^{|I|-1}}(y)=\frac{\gamma_v(y)}{2R\,p_v(0)},\quad y\in\mathcal S_{R^2}.
\end{equation}

\begin{proposition}[Bingham laws as conditioned Gaussian laws]\label{prop:cond-bingham}
Let $N\ge2$.
\begin{enumerate}
\item The uniform measure $\nu_N$ on $S_N$ is $\mu_v$ for $v=(1,\dots,1)\in\R^N$.
If $g$ is a standard Gaussian vector in $\R^N$, then $g/|g|$ is independent of $|g|$, and $\sqrt N\,g/|g|$ has law $\nu_N$.
\item Let $v$ be a variance sequence on $\{1,\dots,N\}$ with $\sum_iv_i=N$, let $z\in\R$, and let $b_i=z-v_i^{-1}$.
Then the probability measure on $S_N$ proportional to $\exp\{\frac12\sum_ib_ix_i^2\}\,\nu_N(\dd x)$ is $\mu_v$.
\item Let $A$ be a real symmetric $N\times N$ matrix with eigenvalues $\alpha_1\ge\dots\ge\alpha_N$, and let $z>\alpha_1$ be its Lagrange multiplier, the unique solution of $N^{-1}\sum_i(z-\alpha_i)^{-1}=1$.
In the eigencoordinates of $A$, the Gibbs measure $\pi_A\propto e^{x^\T Ax/2}\nu_N$ is $\mu_q$, where $q_i=(z-\alpha_i)^{-1}$.
\item For every variance sequence $v$ and $\lambda>0$, if $Y$ has law $\mu_v$, then $\lambda Y$ has law $\mu_{\lambda^2v}$.
Each $\mu_v$ is invariant under the reflections $y_i\mapsto-y_i$, $i\in I$.
\end{enumerate}
\end{proposition}

The last part also holds for countable $v$, with $\mu_v$ as in \cref{def:cond-countable} below; see the remark after that definition.

\begin{proof}
For the first part, we note that the density in \eqref{gp-pin-definition} with $v_i=1$ equals $e^{-N/2}$ on $S_N$, and $\mu_v$ is normalized surface measure, that is, $\nu_N$.
In polar coordinates $g=r\theta$, the density of $g$ is $(2\pi)^{-N/2}e^{-r^2/2}r^{N-1}$ times the surface measure $\mathcal H^{N-1}(\dd\theta)$ on the unit sphere.
This density is a product of a function of $r$ and a function of $\theta$.
It follows that $|g|$ and $g/|g|$ are independent, and that $g/|g|$ is uniformly distributed on the unit sphere.
For the second part, on $S_N$ we have $\frac12\sum_ib_ix_i^2=\frac{zN}2-\frac12\sum_ix_i^2/v_i$.
The first term is constant, and $\mu_v$ lives on $S_N$ because $\sum_iv_i=N$, so the two densities agree up to a constant factor.
We obtain the third part as the case $v_i=q_i$ and $b_i=\alpha_i$ of the second, since $\sum_iq_i=N$ by the definition of $z$.
The existence and uniqueness of $z$ follow since $z\mapsto N^{-1}\sum_i(z-\alpha_i)^{-1}$ decreases continuously from $+\infty$ to $0$ on $(\alpha_1,\infty)$; see \cref{sec:spectral}.
For the fourth part, the map $y\mapsto\lambda y$ carries $\mathcal S_{R^2}$ onto $\mathcal S_{\lambda^2R^2}$ and multiplies surface measure by a constant, and $\sum_i(\lambda y_i)^2/(\lambda^2v_i)=\sum_iy_i^2/v_i$.
The density in \eqref{gp-pin-definition} and the sphere are invariant under each reflection.
\end{proof}

The spherical constraint goes back to the spherical model of \cite[Abstract]{BerlinKac1952}, and its spin-glass version was introduced in \cite[Abstract]{KosterlitzThoulessJones1976}.
In the present normalization, the Lagrange multiplier $z$ is the unique critical point on $(\alpha_1,\infty)$ of
\[
\mathcal A_N(u)=\frac{Nu}2-\frac12\sum_i\log(u-\alpha_i),
\]
since $\mathcal A_N'(u)=\frac12[N-\sum_i(u-\alpha_i)^{-1}]$ is increasing and vanishes exactly at $u=z$.

\section{Countable sequences and coordinate marginals}\label{sec:cond-countable}

For countable $v$ there is no sphere of finite radius when $\sum_iv_i=\infty$, which is the case in the critical window.
We define the conditioned law through its finite-coordinate marginals, following \cite[Section~2.2]{DuHuang2026}, where this is done for sequences with the decay of the inverse Airy gaps; the arguments apply to every variance sequence.
For finite $A\subset I$ such that $I\setminus A$ has at least three elements, we define
\begin{equation}\label{eq:cond-marginal-density}
r_{v,A}(z)=\frac{p_{v,I\setminus A}\big(-U_A(z)\big)}{p_v(0)},\qquad z\in\R^A .
\end{equation}
This is well defined by \cref{lem:cond-three}, since $p_v(0)>0$.

\begin{lemma}[Consistent marginals]\label{lem:cond-consistent}
Let $v$ be a countable variance sequence.
For every finite $A\subset I$, $r_{v,A}$ is a probability density with respect to $\gamma_A$.
If $A\subset A'$ are finite, then the image of $r_{v,A'}\dd\gamma_{A'}$ under the projection $\R^{A'}\to\R^A$ is $r_{v,A}\dd\gamma_A$.
There exists a unique probability measure on $\R^I$ whose image under the projection to $\R^A$ is $r_{v,A}\dd\gamma_A$ for every finite $A\subset I$.
\end{lemma}

\begin{proof}
By \eqref{eq:cond-convolution} at $t=0$, we have $\int r_{v,A}\dd\gamma_A=\E\,p_{v,I\setminus A}(-T_{v,A})/p_v(0)=1$.
It suffices to check consistency when $A'=A\cup\{j\}$.
For fixed $z\in\R^A$, integrating out the coordinate $j$ and applying \eqref{eq:cond-convolution} to the sequence $v|_{I\setminus A}$, split into $\{j\}$ and $I\setminus A'$, we have
\[
\int_{\R}p_{v,I\setminus A'}\big(-U_A(z)-v_j(x^2-1)\big)\phi(x)\dd x=p_{v,I\setminus A}\big(-U_A(z)\big).
\]
Dividing by $p_v(0)$ proves the consistency.
The last statement follows from Kolmogorov's extension theorem.
\end{proof}

\begin{definition}[Countable conditioned law]\label{def:cond-countable}
Let $v$ be a countable variance sequence, and let $Z=(Z_i)_{i\in I}$ have the law constructed in \cref{lem:cond-consistent}.
The \emph{Gaussian law $\mu_v$ conditioned on $T_v=0$} is the law of $Y=(\sqrt{v_i}\,Z_i)_{i\in I}$ on $\R^I$.
\end{definition}

By \eqref{eq:cond-marginal-density}, the laws $\mu_v^{Z_A}$ are invariant under each reflection $z_i\mapsto-z_i$, and the standardized coordinates of $\lambda Y$ with respect to $\lambda^2v$ are again $Z$.
This proves the fourth part of \cref{prop:cond-bingham} for countable $v$.
As shown in \cref{prop:cond-constraint,prop:cond-continuity} below, $\mu_v$ is supported on the set where $\sum_i(y_i^2-v_i)=0$, and, for any enumeration $(i_k)_{k\ge1}$ of $I$, it is the limit of the finite laws $\mu_{v|_{\{i_1,\dots,i_n\}}}$ as $n\to\infty$, in total variation on every finite set of coordinates.

\begin{lemma}[Marginal densities]\label{gp-head-density}
Let $v$ be a variance sequence, and let $A\subset I$.
Suppose that $I\setminus A$ has at least three elements, or that $A$ is infinite and $T_{v,I\setminus A}$ has a bounded density $p_{v,I\setminus A}$.
Then $\mu_v^{Z_A}$ has density
\begin{equation}\label{gp-head-likelihood}
r_{v,A}(z)=\frac{p_{v,I\setminus A}\big(-U_A(z)\big)}{p_v(0)}
\end{equation}
with respect to $\gamma_A$.
If $A$ is infinite, $U_A(z)$ is the series defined for $\gamma_A$-almost every $z$ after \eqref{gp-head-quantities}, and the right side of \eqref{gp-head-likelihood} does not depend on the choice of a version of $p_{v,I\setminus A}$, since $U_A$ has an absolutely continuous law under $\gamma_A$.
\end{lemma}

\begin{proof}
Let $I$ be finite, and let $F$ be a bounded continuous function of $z\in\R^A$.
Define the bounded continuous physical-coordinate test $\widetilde F(y)=F((y_i/\sqrt{v_i})_{i\in A})$ on $\R^I$.
By independence of $g_A$ and $g_{I\setminus A}$, for bounded nonnegative measurable $K$,
\[
\begin{split}
\E[\widetilde F(Y)K(T_v)]&=\int_{\R^A}F(z)\phi_A(z)\int_{\R}K\big(U_A(z)+x\big)p_{v,I\setminus A}(x)\dd x\dd z\\
&=\int_{\R}K(t)\Big[\int_{\R^A}F\phi_A\,p_{v,I\setminus A}\big(t-U_A\big)\dd z\Big]\dd t .
\end{split}
\]
The bracket is a continuous function of $t$, by dominated convergence, since $p_{v,I\setminus A}$ is bounded and continuous (\cref{lem:cond-three}).
By \cref{lem:cond-radial}, applied to the positive and negative parts of $\widetilde F$, $p_v[\widetilde F]$ is another density of the same measure, continuous on $(-R^2,\infty)$.
The two densities agree at $t=0$, by continuity.
Dividing by $p_v[1](0)=p_v(0)$ and using the third part of \cref{lem:cond-radial}, we obtain $\mu_v(\widetilde F)=\int F\,r_{v,A}\dd\gamma_A$, that is, $\E_{\mu_v}F(Z_A)=\int F\,r_{v,A}\dd\gamma_A$.
Since bounded continuous functions determine a probability measure on $\R^A$, this proves \eqref{gp-head-likelihood} for finite $I$.

Let $I$ be countable.
For finite $A$, \eqref{gp-head-likelihood} is the definition of $\mu_v$.
Let $A$ be infinite, and let $\lambda_A(\dd z)=p_v(0)^{-1}p_{v,I\setminus A}(-U_A(z))\gamma_A(\dd z)$, where $p_{v,I\setminus A}$ is a bounded density of $T_{v,I\setminus A}$; this holds in both cases of the hypothesis, by \cref{lem:cond-three}.
For every infinite $A'\subset A$, the variable $T_{v,A'}$ has a bounded continuous density $p_{v,A'}$, by \cref{lem:cond-three}, and the convolution $t\mapsto\int p_{v,I\setminus A}(t-x)p_{v,A'}(x)\dd x$ of a bounded and an integrable function is continuous.
It is a density of $T_{v,(I\setminus A)\cup A'}$, which has a continuous density by \cref{lem:cond-three}, and the two agree everywhere:
\begin{equation}\label{eq:cond-convolution-infinite}
p_{v,(I\setminus A)\cup A'}(t)=\E\,p_{v,I\setminus A}\big(t-T_{v,A'}\big)\qquad\text{for every }t\in\R .
\end{equation}
With $A'=A$ and $t=0$, this shows that $\lambda_A$ is a probability measure.
Let $A_0\subset A$ be finite.
Since $U_A=U_{A_0}+U_{A\setminus A_0}$ almost surely and the coordinates are independent under $\gamma_A$, integrating out the coordinates in $A\setminus A_0$ yields the density
\[
\frac{\E\,p_{v,I\setminus A}\big(-U_{A_0}(z)-T_{v,A\setminus A_0}\big)}{p_v(0)}
=\frac{p_{v,I\setminus A_0}\big(-U_{A_0}(z)\big)}{p_v(0)}=r_{v,A_0}(z),\qquad z\in\R^{A_0},
\]
where we used \eqref{eq:cond-convolution-infinite} with $A'=A\setminus A_0$.
We conclude that $\lambda_A$ and $\mu_v^{Z_A}$ have the same finite-dimensional marginals.
Since two probability measures on $\R^A$ that agree on cylinder sets are equal, we have $\lambda_A=\mu_v^{Z_A}$.
\end{proof}

For finite $A$, the formula \eqref{gp-head-likelihood} appears in \cite[Lemma~4.6]{DuHuang2026} for finite sequences and in \cite[(2.14)]{DuHuang2026} for countable ones.

\begin{proposition}[The radial constraint]\label{prop:cond-constraint}
Let $v$ be a countable variance sequence and $(i_k)_{k\ge1}$ an enumeration of $I$.
Under $\mu_v$, the partial sums $\sum_{k\le n}(Y_{i_k}^2-v_{i_k})=\sum_{k\le n}v_{i_k}(Z_{i_k}^2-1)$ converge to zero almost surely and in $L^2(\mu_v)$.
\end{proposition}

\begin{proof}
This is proved in \cite[Lemma~2.4]{DuHuang2026} for sequences with the decay of the inverse Airy gaps; we give the argument for general $v$.
Let $A_n=\{i_1,\dots,i_n\}$ and $Q_n=U_{A_n}(Z_{A_n})$.
For $n\ge3$, the density $q_n$ of $T_{v,A_n}$ under the Gaussian product law satisfies $\|q_n\|_\infty\le\|p_{v,A_3}\|_\infty$, since $T_{v,A_n}$ is the sum of $T_{v,A_3}$ and an independent variable.
By \cref{gp-head-density}, we have
\[
\E_{\mu_v}Q_n^2=\frac{1}{p_v(0)}\int_{\R}x^2q_n(x)\,p_{v,I\setminus A_n}(-x)\dd x
\le\frac{\|p_{v,A_3}\|_\infty}{p_v(0)}\,\E\,T_{v,I\setminus A_n}^2
=\frac{\|p_{v,A_3}\|_\infty}{p_v(0)}\,2\sum_{k>n}v_{i_k}^2,
\]
which tends to zero.
For almost sure convergence, let $B=A_3$.
Applying \cref{gp-head-density} with $A=I\setminus B$, we find that the law of $Z_{I\setminus B}$ under $\mu_v$ is absolutely continuous with respect to $\gamma_{I\setminus B}$.
Under $\gamma_{I\setminus B}$, the partial sums $\sum_{3<k\le n}v_{i_k}(z_{i_k}^2-1)$ converge almost surely, as noted after \eqref{gp-standardization}.
It follows that $Q_n$ converges $\mu_v$-almost surely, and its limit is zero because $Q_n\to0$ in $L^2(\mu_v)$.
\end{proof}

The next theorem compares a set of coordinates of $\mu_v$ with independent Gaussian variables when the coordinates carry at most a fixed fraction of the variance of $T_v$.

\begin{theorem}[Gaussian approximation of coordinate marginals]\label{gp-growing-head}
There exist universal constants $\eta_*>0$ and $C<\infty$ with the following property.
Let $v$ be a variance sequence with $\eta_v\le\eta_*$, and let $A\subset I$ with $h_A\le1/2$.
Then $I\setminus A$ has at least three elements, and
\begin{equation}\label{gp-growing-head-bound}
\big\|\mu_v^{Z_A}-\gamma_A\big\|_{\TV}\le C(\eta_v+h_A),\qquad 0\le r_{v,A}\le C .
\end{equation}
Further, for each $\eps\in(0,1)$ there exist $\eta(\eps)>0$ and $C_\eps<\infty$ such that $\eta_v\le\eta(\eps)$ and $h_A\le1-\eps$ imply that $I\setminus A$ has at least three elements and $r_{v,A}\le C_\eps$.
\end{theorem}

For the uniform measure, Borel gave the Gaussian limit of one coordinate; see \cite[Section~6.1]{DiaconisFreedman1987}. Stam proved a bound in total variation for a number of coordinates that grows more slowly than $\sqrt N$ \cite{Stam1982}, and Diaconis and Freedman proved a bound of order $k/N$ for $k$ coordinates, which is small whenever $k=o(N)$ \cite[(1)]{DiaconisFreedman1987}; see \cite[Sections~6.1 and~6.4]{DiaconisFreedman1987} for the history.
\Cref{gp-growing-head} allows arbitrary variances, with constants that do not depend on the smallest variance.

\begin{proof}
The standardized variances of $v|_{I\setminus A}$ are $v_i/S_{v|_{I\setminus A}}=\hat v_i(1-h_A)^{-1/2}$, and in particular $\eta_{v|_{I\setminus A}}\le\eta_v(1-h_A)^{-1/2}$.
Let $\eta_{\mathrm{loc}}$, $c_0$, $C_0$ be as in \cref{gp-local-limit}.
If $\eta_v(1-h_A)^{-1/2}\le\eta_{\mathrm{loc}}$, then $I\setminus A$ has at least $(2\eta_{\mathrm{loc}}^2)^{-1}\ge3$ elements, and \cref{gp-head-density} applies.
Writing $p_{v,I\setminus A}(x)=S_{v|_{I\setminus A}}^{-1}f_{v|_{I\setminus A}}(x/S_{v|_{I\setminus A}})$ and $p_v(0)=S_v^{-1}f_v(0)$ in \eqref{gp-head-likelihood}, we obtain
\begin{equation}\label{eq:cond-head-ratio}
r_{v,A}(z)=\frac{1}{(1-h_A)^{1/2}}\,\frac{f_{v|_{I\setminus A}}(-\tau_A)}{f_v(0)},\qquad
\tau_A=\frac{U_A(z)}{S_v(1-h_A)^{1/2}} .
\end{equation}
By \cref{gp-local-limit} with $j=0$, we have $f_{v|_{I\setminus A}}=\phi+e_1$ and $f_v(0)=\phi(0)+e_2$, where
\[
\|e_1\|_\infty\le C_0\eta_v(1-h_A)^{-1/2},\qquad |e_2|\le C_0\eta_v .
\]

Let $h_A\le1/2$ and $\eta_*\le\eta_{\mathrm{loc}}/\sqrt2$, with $\sqrt2C_0\eta_*\le\phi(0)/2$.
Then $f_v(0)\ge\phi(0)/2$ and $f_{v|_{I\setminus A}}\le\phi(0)+\sqrt2C_0\eta_v$, and \eqref{eq:cond-head-ratio} gives $r_{v,A}\le C$.
Further, we have $|(1-h)^{-1/2}-1|\le h$ for $h\in[0,1/2]$ and $|\phi(x)-\phi(0)|\le\phi(0)x^2/2$.
Inserting these bounds into the numerator of $r_{v,A}-1=[(1-h_A)^{-1/2}f_{v|_{I\setminus A}}(-\tau_A)-f_v(0)]/f_v(0)$, we find that it is at most $C(\eta_v+h_A+\tau_A^2)$ in absolute value.
We conclude that
\[
|r_{v,A}(z)-1|\le C\big(\eta_v+h_A+\tau_A(z)^2\big).
\]
Under $\gamma_A$, we have $\E\,U_A^2=2\sum_{i\in A}v_i^2=h_AS_v^2$, which gives $\int\tau_A^2\dd\gamma_A=h_A/(1-h_A)\le2h_A$.
Recalling that $\|\mu_v^{Z_A}-\gamma_A\|_{\TV}=\frac12\int|r_{v,A}-1|\dd\gamma_A$, we conclude that \eqref{gp-growing-head-bound} holds.

Now let $h_A\le1-\eps$, and let $\eta(\eps)=\min\{\eta_{\mathrm{loc}}\eps^{1/2},\phi(0)/(2C_0)\}$.
Then $\eta_{v|_{I\setminus A}}\le\eta_v\eps^{-1/2}\le\eta_{\mathrm{loc}}$ and $f_v(0)\ge\phi(0)/2$, so that \eqref{eq:cond-head-ratio} yields $r_{v,A}\le\eps^{-1/2}(\phi(0)+C_0\eta_v\eps^{-1/2})/(\phi(0)/2)\le C_\eps$.
\end{proof}

\section{Moments of coordinate marginals}\label{sec:cond-moments}

The density bounds of \cref{gp-growing-head} transfer moments of Gaussian observables to the conditioned law.

\begin{theorem}[Moments under the conditioned law]\label{gp-moments}
Let $\eta_*\le1/2$ be as in \cref{gp-growing-head}, and let $v$ be a variance sequence with $\eta_v\le\eta_*$.
If $A\subset I$ satisfies $h_A\le1/2$, $p>1$, and $F\in L^p(\gamma_A)$, then $F(Z_A)$ is integrable under $\mu_v$ and
\begin{equation}\label{gp-head-moment-comparison}
\big|\E_{\mu_v}F(Z_A)-\E_{\gamma_A}F\big|\le C_p\|F\|_{L^p(\gamma_A)}(\eta_v+h_A)^{1-1/p}.
\end{equation}
In particular, uniformly in $i\in I$,
\begin{equation}\label{gp-second-fourth}
\begin{aligned}
\E_{\mu_v}Y_i^2&=v_i\big[1+O(\eta_v^{1/2})\big],\\
\E_{\mu_v}Y_i^4&=3v_i^2\big[1+O(\eta_v^{1/2})\big],\\
\Var_{\mu_v}(Y_i^2)&=2v_i^2\big[1+O(\eta_v^{1/2})\big],
\end{aligned}
\end{equation}
where the implicit constants are universal.
For each $\eps\in(0,1)$, let $\eta(\eps)$ and $C_\eps$ be as in \cref{gp-growing-head}.
If $\eta_v\le\eta(\eps)$ and $h_A\le1-\eps$, then for every $p>1$, $F\in L^p(\gamma_A)$, and $L>0$,
\begin{equation}\label{eq:cond-ui}
\E_{\mu_v}\big[|F(Z_A)|\ind\{|F(Z_A)|>L\}\big]\le C_\eps L^{1-p}\|F\|_{L^p(\gamma_A)}^p .
\end{equation}
In particular, every family of such observables $F(Z_A)$ with bounded $L^p(\gamma_A)$ norms is uniformly integrable under the corresponding conditioned laws.
\end{theorem}

\begin{proof}
For \eqref{gp-head-moment-comparison}, let $q=p/(p-1)$.
We have $\|r_{v,A}-1\|_\infty\le C$ and $\int|r_{v,A}-1|\dd\gamma_A\le C(\eta_v+h_A)$ by \cref{gp-growing-head}, so
\[
\|r_{v,A}-1\|_{L^q(\gamma_A)}\le\|r_{v,A}-1\|_\infty^{1-1/q}\|r_{v,A}-1\|_{L^1(\gamma_A)}^{1/q}\le C_p(\eta_v+h_A)^{1-1/p}.
\]
The bound $r_{v,A}\le C$ implies that $F(Z_A)$ is integrable under $\mu_v$.
By \cref{gp-head-density} and H\"older's inequality,
\[
\big|\E_{\mu_v}F(Z_A)-\E_{\gamma_A}F\big|
=\Big|\int F(r_{v,A}-1)\dd\gamma_A\Big|
\le\|F\|_{L^p(\gamma_A)}\|r_{v,A}-1\|_{L^q(\gamma_A)},
\]
which proves \eqref{gp-head-moment-comparison}.
For $A=\{i\}$ we have $h_A=2\hat v_i^2\le2\eta_v^2\le1/2$.
Taking $p=2$ and $F(z)=z_i^2$ or $F(z)=z_i^4$, whose Gaussian $L^2$ norms are universal constants, gives
\[
\big|\E_{\mu_v}Z_i^2-1\big|+\big|\E_{\mu_v}Z_i^4-3\big|
\le C(\eta_v+2\eta_v^2)^{1/2}\le C\eta_v^{1/2}.
\]
Scaling by $v_i$ and $v_i^2$ gives the first two estimates in \eqref{gp-second-fourth}; the third follows by subtracting $(\E_{\mu_v}Y_i^2)^2$ from $\E_{\mu_v}Y_i^4$.
Finally, under the assumptions of the last statement, $r_{v,A}\le C_\eps$ by \cref{gp-growing-head}, and
\[
\E_{\mu_v}\big[|F(Z_A)|\ind\{|F(Z_A)|>L\}\big]\le C_\eps\int|F|\ind\{|F|>L\}\dd\gamma_A\le C_\eps L^{1-p}\int|F|^p\dd\gamma_A,
\]
which is \eqref{eq:cond-ui}.
\end{proof}

\section{Exponential tilts and quadratic statistics}\label{sec:cond-quadratic}

\begin{lemma}[Quadratic exponential moments]\label{gp-real-tilt}
Let $v$ be a variance sequence, and let $w=(w_i)_{i\in I}$ be real numbers, with only finitely many nonzero when $I$ is countable.
Let $t\in\R$ satisfy $1-tw_i>0$ for every $i$, and let
\[
v_i(t)=\frac{v_i}{1-tw_i},\qquad \Delta(t)=\sum_{i\in I}\big(v_i(t)-v_i\big).
\]
Then $v(t)$ is a variance sequence, and
\begin{equation}\label{gp-real-tilt-identity}
\E_{\mu_v}\exp\Big\{\frac t2\sum_{i\in I}w_iZ_i^2\Big\}
=\prod_{i\in I}(1-tw_i)^{-1/2}\,\frac{p_{v(t)}\big(-\Delta(t)\big)}{p_v(0)} .
\end{equation}
\end{lemma}

All the quantities on the right side are real, and no smallness of $\Delta(t)$ is assumed.

\begin{proof}
For one coordinate, multiplying the Gaussian density with variance $v_i$ by $\exp\{tw_iy^2/(2v_i)\}$ gives $(1-tw_i)^{-1/2}$ times the Gaussian density with variance $v_i(t)$.
Multiplying over $i$, we have, with $\gamma_v$ as in \cref{lem:cond-radial},
\begin{equation}\label{eq:cond-tilt-density}
\exp\Big\{\frac t2\sum_iw_i\frac{y_i^2}{v_i}\Big\}\gamma_v(y)=\prod_i(1-tw_i)^{-1/2}\,\gamma_{v(t)}(y).
\end{equation}
Let $I$ be finite, $R^2=\sum_iv_i$, and let $W(y)$ be the exponential on the left of \eqref{eq:cond-tilt-density}.
Since $\sum_iv_i(t)=R^2+\Delta(t)$, the definition \eqref{eq:cond-weighted-density} applied to $v(t)$ at the point $s-\Delta(t)$ integrates over the same sphere $\mathcal S_{R^2+s}$ as $p_v[W](s)$, and \eqref{eq:cond-tilt-density} implies that
\[
p_v[W](s)=\prod_i(1-tw_i)^{-1/2}\,p_{v(t)}\big(s-\Delta(t)\big),\qquad s>-R^2 .
\]
Evaluating at $s=0$ and dividing by $p_v(0)$, we obtain \eqref{gp-real-tilt-identity} from the third part of \cref{lem:cond-radial}.

Let $I$ be countable, and let $F\subset I$ be a finite set containing the indices with $w_i\neq0$.
The sequence $v(t)$ agrees with $v$ off $F$.
In particular, it is a variance sequence and $p_{v(t),I\setminus F}=p_{v,I\setminus F}$.
By \cref{gp-head-density} and \eqref{eq:cond-tilt-density},
\[
\E_{\mu_v}\exp\Big\{\frac t2\sum_{i\in F}w_iZ_i^2\Big\}
=\prod_{i\in F}(1-tw_i)^{-1/2}\,\frac{\E\,p_{v,I\setminus F}\big(-\sum_{i\in F}(v_i(t)g_i^2-v_i)\big)}{p_v(0)} .
\]
Since $\sum_{i\in F}(v_i(t)g_i^2-v_i)=T_{v(t),F}+\Delta(t)$, the expectation equals $p_{v(t)}(-\Delta(t))$ by \eqref{eq:cond-convolution} for $v(t)$.
This proves \eqref{gp-real-tilt-identity}.
\end{proof}

For independent Gaussian variables, the analogous moment bound for quadratic statistics is classical; we record it with its short proof, since it is also used in \cref{prop:cond-continuity}.

\begin{lemma}[Quadratic statistics of independent Gaussians]\label{lem:cond-chaos}
For every $p\ge1$ there exists $C_p<\infty$ such that, for every finite or countable index set $J$ and every real $w\in\ell^2(J)$, the series $X=\sum_{i\in J}w_i(g_i^2-1)$ satisfies $\E e^{tX}\le e^{2t^2\|w\|_2^2}$ for $|t|\le(4\|w\|_2)^{-1}$, and $\|X\|_{L^p}\le C_p\|w\|_2$. Here $(4\|w\|_2)^{-1}=+\infty$ when $w=0$.
\end{lemma}

\begin{proof}
If $w=0$, then $X=0$ and both claims hold immediately. Fix a finite nonzero $w$ and $t$ with $|t|\le(4\|w\|_2)^{-1}$, so that $|2tw_i|\le1/2$ for every $i$.
For $|2u|\le1/2$, we have $-u-\frac12\log(1-2u)=\sum_{k\ge2}(2u)^k/(2k)\le\frac14(2u)^2/(1-|2u|)\le2u^2$.
Since $\E e^{sw(g^2-1)}=e^{-sw}(1-2sw)^{-1/2}$ for $2sw<1$, we have $\log\E e^{tX}=\sum_i[-tw_i-\frac12\log(1-2tw_i)]\le2t^2\|w\|_2^2$.
For countable $J$, the partial sums converge almost surely along a subsequence, and the same bound follows from Fatou's lemma.
At $t=\pm(4\|w\|_2)^{-1}$, Markov's inequality implies that $\PP(|X|>s\|w\|_2)\le2e^{1/8}e^{-s/4}$ for $s\ge0$.
Integrating this tail against $ps^{p-1}\dd s$ yields $\E|X|^p\le C_p^p\|w\|_2^p$.
\end{proof}

\begin{theorem}[Quadratic statistics under the conditioned law]\label{gp-quadratic-lp}
There exist universal constants $\eta_0,c_*>0$ and $C<\infty$ such that the following holds for every variance sequence $v$ with $\eta_v\le\eta_0$.
Let $w=(w_i)_{i\in I}$ be real numbers, with only finitely many nonzero, and let
\[
Q_w=\sum_{i\in I}w_i\big(Z_i^2-1\big)=\sum_{i\in I}w_i\Big(\frac{Y_i^2}{v_i}-1\Big),\qquad \|w\|_2=\Big(\sum_iw_i^2\Big)^{1/2}>0 .
\]
Then
\begin{equation}\label{gp-quadratic-exponential-bound}
\E_{\mu_v}e^{tQ_w}\le Ce^{Ct^2\|w\|_2^2},\qquad |t|\le c_*/\|w\|_2 .
\end{equation}
For every real $p\ge1$, there exists $C_p<\infty$ such that
\begin{equation}\label{gp-quadratic-lp-bound}
\|Q_w\|_{L^p(\mu_v)}\le C_p\|w\|_2 .
\end{equation}
If $I$ is countable and $w\in\ell^2(I)$, then the series defining $Q_w$ converges in $L^p(\mu_v)$ for every $p\ge1$, and its limit satisfies the same two bounds.
\end{theorem}

If $w=0$, then $Q_w=0$ and there is nothing to prove.
The theorem bounds $Q_w$ itself, which is centered at the mean under the independent Gaussian law; the mean $\E_{\mu_v}Q_w$ need not vanish.

\begin{proof}
We set $c_*=1/8$ and fix $t$ with $|t|\le c_*/\|w\|_2$, so that $|2tw_i|\le1/4$ for every $i$.
We apply \cref{gp-real-tilt} to the coefficients $2w_i$, with
\[
v_i(t)=\frac{v_i}{1-2tw_i}\in\Big[\frac45v_i,\frac43v_i\Big],\qquad
\frac45S_v\le S_{v(t)}\le\frac43S_v,\qquad
\eta_{v(t)}\le\frac53\eta_v .
\]
Since $tQ_w=t\sum_iw_iZ_i^2-t\sum_iw_i$, we find from \eqref{gp-real-tilt-identity} that
\[
\E_{\mu_v}e^{tQ_w}=\exp\Big\{\sum_{i\in I}\Big[-tw_i-\frac12\log(1-2tw_i)\Big]\Big\}\,\frac{p_{v(t)}\big(-\Delta(t)\big)}{p_v(0)} .
\]
Let $\eta_0=\frac35\eta_{\mathrm{loc}}$, so that \eqref{gp-local-limit-consequences} applies to $v$ and $v(t)$.
Using the upper bound for $f_{v(t)}$ and the lower bound for $f_v(0)$, we have
\[
\frac{p_{v(t)}(-\Delta(t))}{p_v(0)}\le\frac{C/S_{v(t)}}{c_0/S_v}\le C,
\]
whatever the size of $\Delta(t)$.
The elementary inequality in the proof of \cref{lem:cond-chaos} bounds the exponent by $2t^2\|w\|_2^2$, which proves \eqref{gp-quadratic-exponential-bound}.

At $t=\pm c_*/\|w\|_2$, Markov's inequality and \eqref{gp-quadratic-exponential-bound} give $\mu_v(|Q_w|>s\|w\|_2)\le Ce^{-c_*s}$ for $s\ge0$.
Integrating this tail against $ps^{p-1}\dd s$ gives \eqref{gp-quadratic-lp-bound}.
For countable $I$ and $w\in\ell^2(I)$, the bound \eqref{gp-quadratic-lp-bound} applied to differences of partial sums shows that the partial sums form a Cauchy sequence in every $L^p(\mu_v)$.
The limits for different $p$ agree almost surely, since they are also limits in probability.
Along an almost surely convergent subsequence, Fatou's lemma yields \eqref{gp-quadratic-exponential-bound} for the limit, since $c_*/\|w\|_2$ is at most the corresponding threshold for each partial sum; norm convergence yields \eqref{gp-quadratic-lp-bound}.
\end{proof}

The case $p=2$ is used often enough to be stated separately, in the variables $Y_i$.
Its left side is a second moment about the deterministic center $\sum_iw_iv_i$, and it bounds the variance of $\sum_iw_iY_i^2$ under $\mu_v$.

\begin{corollary}[Quadratic second-moment bound]\label{gp-quadratic-variance}
Let $\eta_0$ be the constant of \cref{gp-quadratic-lp}.
There exists a universal constant $C$ such that, for every variance sequence $v$ with $\eta_v\le\eta_0$ and all real $w_i$, only finitely many nonzero,
\begin{equation}\label{gp-quadratic-variance-bound}
\E_{\mu_v}\Big[\sum_{i\in I}w_i\big(Y_i^2-v_i\big)\Big]^2\le C\sum_{i\in I}w_i^2v_i^2 .
\end{equation}
For countable $I$ and $\sum_iw_i^2v_i^2<\infty$, the series is the $L^2(\mu_v)$ limit of its partial sums, and \eqref{gp-quadratic-variance-bound} holds.
\end{corollary}

\begin{proof}
We have $\sum_iw_i(Y_i^2-v_i)=Q_{w'}$ with $w_i'=w_iv_i$, and the claim is \cref{gp-quadratic-lp} with $p=2$.
\end{proof}

\section{Continuity in the variance sequence}\label{sec:cond-continuity}

The estimates of the previous sections assume that $\eta_v$ is small.
The conditioned law in the critical window has a fixed positive $\eta_v$. Instead, we use the following domination by independent Gaussian variables, together with continuity of the law under convergence of the variance sequence in $\ell^2$.
For finite $B\subset I$ such that $B$ and $I\setminus B$ both have at least three elements, let
\begin{equation}\label{eq:cond-domination-constant}
M(v,B)=\frac{\max\{\|p_{v,B}\|_\infty,\|p_{v,I\setminus B}\|_\infty\}}{p_v(0)} .
\end{equation}

\begin{proposition}[Continuity and domination]\label{prop:cond-continuity}
Let $v$ be a variance sequence.
\begin{enumerate}
\item Let $B\subset I$ be finite, and suppose that $B$ and $I\setminus B$ both have at least three elements.
Then for all nonnegative measurable functions $H_1$ on $\R^B$ and $H_2$ on $\R^{I\setminus B}$,
\begin{equation}\label{eq:cond-domination}
\E_{\mu_v}H_1(Z_B)\le M(v,B)\int H_1\dd\gamma_B,\qquad
\E_{\mu_v}H_2(Z_{I\setminus B})\le M(v,B)\int H_2\dd\gamma_{I\setminus B}.
\end{equation}
In particular, for every $p\ge1$ we have $\sup_{i\in I}\E_{\mu_v}|Z_i|^p\le M(v,B)\,\E|G|^p$, and $\E_{\mu_v}\sum_iu_iZ_i^2\le M(v,B)\sum_iu_i$ for all $u_i\ge0$.
For every real $w\in\ell^2(I)$, the series $\sum_iw_i(Z_i^2-1)$ converges in $L^p(\mu_v)$, and
\begin{equation}\label{eq:cond-chaos-bound}
\Big\|\sum_{i\in I}w_i\big(Z_i^2-1\big)\Big\|_{L^p(\mu_v)}\le2C_pM(v,B)^{1/p}\|w\|_2,
\end{equation}
where $C_p$ is the constant of \cref{lem:cond-chaos}.
\item Let $v^{(n)}$, $n\ge1$, be variance sequences indexed by subsets $I_n\subset I$, extended by zero to $I$, such that $\sum_{i\in I}(v_i^{(n)}-v_i)^2\to0$.
Suppose that $I$ has at least three elements.
Then $p_{v^{(n)}}(0)\to p_v(0)$.
Let $A\subset I$ be finite such that $I\setminus A$ has at least three elements.
Then $A\subset I_n$ and $I_n\setminus A$ has at least three elements for all large $n$, and
\[
\begin{gathered}
\sup_{x\in\R}\big|p_{v^{(n)},I_n\setminus A}(x)-p_{v,I\setminus A}(x)\big|\to0,\\
\big\|\mu_{v^{(n)}}^{Z_A}-\mu_v^{Z_A}\big\|_{\TV}\to0,\qquad
\big\|\mu_{v^{(n)}}^{Y_A}-\mu_v^{Y_A}\big\|_{\TV}\to0,
\end{gathered}
\]
where $\mu^{Y_A}$ is the law of $Y_A=(Y_i)_{i\in A}$ under $\mu$.
If $B$ is as in the first part, then $M(v^{(n)},B)\to M(v,B)$.
In particular, there exists $n_0$ such that the bounds of the first part hold for $\mu_{v^{(n)}}$, $n\ge n_0$, with $M(v,B)$ replaced by $2M(v,B)$.
\end{enumerate}
\end{proposition}

For variance sequences with the decay of the inverse Airy gaps, the convergence of finite-coordinate marginals in the second part, with uniform bounds on second moments, is proved in \cite[Lemma~4.7]{DuHuang2026}; see (4.24)--(4.27) there.
The domination in the first part, from which we obtain all moments and the bound \eqref{eq:cond-chaos-bound}, is the additional input needed in \cref{sec:crit-process}.

\begin{proof}
For the first part, we use \cref{gp-head-density} with $A=B$ and with $A=I\setminus B$.
The densities $r_{v,B}=p_{v,I\setminus B}(-U_B)/p_v(0)$ and $r_{v,I\setminus B}=p_{v,B}(-U_{I\setminus B})/p_v(0)$ are bounded by $M(v,B)$, which proves \eqref{eq:cond-domination}.
We obtain the bound on $\E_{\mu_v}|Z_i|^p$ by applying \eqref{eq:cond-domination} to $H(z)=|z_i|^p$ in whichever of the two sets contains $i$, and the bound on $\E_{\mu_v}\sum_iu_iZ_i^2$ follows by monotone convergence.
For \eqref{eq:cond-chaos-bound}, the partial sums of $\sum_{i\in I\setminus B}w_i(Z_i^2-1)$, along any enumeration, form a Cauchy sequence in $L^p(\mu_v)$, since by \eqref{eq:cond-domination} and \cref{lem:cond-chaos},
\[
\E_{\mu_v}\Big|\sum_{i\in F}w_i\big(Z_i^2-1\big)\Big|^p\le M(v,B)\,C_p^p\Big(\sum_{i\in F}w_i^2\Big)^{p/2}
\]
for every finite $F\subset I\setminus B$.
The same bound holds for the limit, with $F$ replaced by $I\setminus B$, and for the sum over $B$.
The triangle inequality and $\|w|_B\|_2+\|w|_{I\setminus B}\|_2\le2\|w\|_2$ prove \eqref{eq:cond-chaos-bound}.

For the second part, we take the same independent variables $g$ for all $n$, so that $T_{v^{(n)}}=\sum_{i\in I}v_i^{(n)}(g_i^2-1)$.
Then $\E|T_{v^{(n)}}-T_v|^2=2\sum_i(v_i^{(n)}-v_i)^2\to0$, and the characteristic functions $\varphi_n$ of $T_{v^{(n)}}$ converge pointwise to the characteristic function $\varphi$ of $T_v$.
Fix three indices $j_1,j_2,j_3\in I$, and let $m=\frac12\min_kv_{j_k}$.
For large $n$, we have $v_{j_k}^{(n)}\ge m$, and the argument in the proof of \cref{lem:cond-three} shows that $|\varphi_n(u)|+|\varphi(u)|\le2(1+4m^2u^2)^{-3/4}$.
By Fourier inversion and dominated convergence,
\[
\sup_x\big|p_{v^{(n)}}(x)-p_v(x)\big|\le\frac1{2\pi}\int_{\R}|\varphi_n(u)-\varphi(u)|\dd u\to0 .
\]
The same argument, with three indices of $I\setminus A$, proves the uniform convergence of $p_{v^{(n)},I_n\setminus A}$.
Every finite subset of $I$ is contained in $I_n$ for large $n$, since $v_i^{(n)}\to v_i>0$ for each $i$; this proves the claims about $A\subset I_n$.
The laws $\mu_{v^{(n)}}^{Z_A}$ and $\mu_v^{Z_A}$ have the densities $r_{v^{(n)},A}$ and $r_{v,A}$ with respect to $\gamma_A$, by \cref{gp-head-density}.
Using $U_A^{(n)}(z)=\sum_{i\in A}v^{(n)}_i(z_i^2-1)\to U_A(z)$, the uniform convergence of densities, the continuity of $p_{v,I\setminus A}$, and $p_{v^{(n)}}(0)\to p_v(0)>0$, we find that $r_{v^{(n)},A}\to r_{v,A}$ pointwise.
By Scheff\'e's lemma, we have $\int|r_{v^{(n)},A}-r_{v,A}|\dd\gamma_A\to0$, which is the convergence of $\mu_{v^{(n)}}^{Z_A}$ in total variation.

For $Y_A$, let $D_n$ and $D$ be the diagonal matrices with entries $(v_i^{(n)})^{1/2}$ and $v_i^{1/2}$, $i\in A$.
Total variation is invariant under the bijection $D_n$, and by the triangle inequality we have
\[
\big\|\mu_{v^{(n)}}^{Y_A}-\mu_v^{Y_A}\big\|_{\TV}
\le\big\|\mu_{v^{(n)}}^{Z_A}-\mu_v^{Z_A}\big\|_{\TV}+\big\|\Law(D_nD^{-1}Y_A)-\Law(Y_A)\big\|_{\TV},
\]
where $Y_A$ has law $\mu_v^{Y_A}$.
The law of $Y_A$ has a Lebesgue density $\rho\in L^1(\R^A)$, and the law of $D_nD^{-1}Y_A$ has the density $\rho(DD_n^{-1}y)\det(DD_n^{-1})$.
Since $D_nD^{-1}\to\mathrm{Id}$, this density converges to $\rho$ in $L^1$: this is clear for continuous $\rho$ with compact support, by dominated convergence, and the general case follows by approximation, since the change of variables preserves the $L^1$ norm.
This proves the convergence of $\mu_{v^{(n)}}^{Y_A}$.
Finally, the same Fourier argument shows that $p_{v^{(n)},B}\to p_{v,B}$ and $p_{v^{(n)},I_n\setminus B}\to p_{v,I\setminus B}$ uniformly, so that their suprema converge, and together with $p_{v^{(n)}}(0)\to p_v(0)$ this gives $M(v^{(n)},B)\to M(v,B)$.
\end{proof}